let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f1, f2 being PartFunc of M, the carrier of V
for f3 being PartFunc of M,COMPLEX holds (f3 (#) f1) - (f3 (#) f2) = f3 (#) (f1 - f2)
let V be ComplexNormSpace; :: thesis: for f1, f2 being PartFunc of M, the carrier of V
for f3 being PartFunc of M,COMPLEX holds (f3 (#) f1) - (f3 (#) f2) = f3 (#) (f1 - f2)
let f1, f2 be PartFunc of M, the carrier of V; :: thesis: for f3 being PartFunc of M,COMPLEX holds (f3 (#) f1) - (f3 (#) f2) = f3 (#) (f1 - f2)
let f3 be PartFunc of M,COMPLEX; :: thesis: (f3 (#) f1) - (f3 (#) f2) = f3 (#) (f1 - f2)
A1: dom ((f3 (#) f1) - (f3 (#) f2)) = (dom (f3 (#) f1)) /\ (dom (f3 (#) f2)) by Def2
.= (dom (f3 (#) f1)) /\ ((dom f3) /\ (dom f2)) by Def3
.= ((dom f3) /\ (dom f1)) /\ ((dom f3) /\ (dom f2)) by Def3
.= ((dom f3) /\ ((dom f3) /\ (dom f1))) /\ (dom f2) by XBOOLE_1:16
.= (((dom f3) /\ (dom f3)) /\ (dom f1)) /\ (dom f2) by XBOOLE_1:16
.= (dom f3) /\ ((dom f1) /\ (dom f2)) by XBOOLE_1:16
.= (dom f3) /\ (dom (f1 - f2)) by Def2
.= dom (f3 (#) (f1 - f2)) by Def3 ;

now

let x be Element of M; :: thesis: ( x in dom (f3 (#) (f1 - f2)) implies (f3 (#) (f1 - f2)) /. x = ((f3 (#) f1) - (f3 (#) f2)) /. x )
assume A2: x in dom (f3 (#) (f1 - f2)) ; :: thesis: (f3 (#) (f1 - f2)) /. x = ((f3 (#) f1) - (f3 (#) f2)) /. x
then x in (dom f3) /\ (dom (f1 - f2)) by Def3;
then A3: x in dom (f1 - f2) by XBOOLE_0:def 4;
A4: x in (dom (f3 (#) f1)) /\ (dom (f3 (#) f2)) by A1, A2, Def2;
then A5: x in dom (f3 (#) f1) by XBOOLE_0:def 4;
A6: x in dom (f3 (#) f2) by A4, XBOOLE_0:def 4;
thus (f3 (#) (f1 - f2)) /. x = (f3 /. x) * ((f1 - f2) /. x) by A2, Def3
.= (f3 /. x) * ((f1 /. x) - (f2 /. x)) by A3, Def2
.= ((f3 /. x) * (f1 /. x)) - ((f3 /. x) * (f2 /. x)) by CLVECT_1:10
.= ((f3 (#) f1) /. x) - ((f3 /. x) * (f2 /. x)) by A5, Def3
.= ((f3 (#) f1) /. x) - ((f3 (#) f2) /. x) by A6, Def3
.= ((f3 (#) f1) - (f3 (#) f2)) /. x by A1, A2, Def2 ; :: thesis: verum

end;

hence (f3 (#) f1) - (f3 (#) f2) = f3 (#) (f1 - f2) by A1, PARTFUN2:3; :: thesis: verum